'What then is Love?' I asked; 'Is he mortal?' 'No.' 'What then?' 'As in the former instance, he is neither mortal nor immortal, but in a mean between the two.' 'What is he, Diotima?' 'He is a great spirit (daimon), and like all spirits he is intermediate between the divine and the mortal.' 'And what,' I said, 'is his power?' 'He interprets,' she replied, 'between gods and men, conveying and taking across to the gods the prayers and sacrifices of men, and to men the commands and replies of the gods; he is the mediator who spans the chasm which divides them, and therefore in him all is bound together, and through him the arts of the prophet and the priest, their sacrifices and mysteries and charms, and all prophecy and incantation, find their way. For God mingles not with man; but through Love all the intercourse and converse of God with man, whether awake or asleep, is carried on. The wisdom which understands this is spiritual; all other wisdom, such as that of arts and handicrafts, is mean and vulgar. Now these spirits or intermediate powers are many and diverse, and one of them is Love.' 'And who,' I said, 'was his father, and who his mother?' 'The tale,' she said, 'will take time; nevertheless I will tell you. On the birthday of Aphrodite there was a feast of the gods, at which the god Poros or Plenty, who is the son of Metis or Discretion, was one of the guests. When the feast was over, Penia or Poverty, as the manner is on such occasions, came about the doors to beg. Now Plenty who was the worse for nectar (there was no wine in those days), went into the garden of Zeus and fell into a heavy sleep, and Poverty considering her own straitened circumstances, plotted to have a child by him, and accordingly she lay down at his side and conceived Love, who partly because he is naturally a lover of the beautiful, and because Aphrodite is herself beautiful, and also because he was born on her birthday, is her follower and attendant. And as his parentage is, so also are his fortunes. In the first place he is always poor, and anything but tender and fair, as the many imagine him; and he is rough and squalid, and has no shoes, nor a house to dwell in; on the bare earth exposed he lies under the open heaven, in the streets, or at the doors of houses, taking his rest; and like his mother he is always in distress. Like his father too, whom he also partly resembles, he is always plotting against the fair and good; he is bold, enterprising, strong, a mighty hunter, always weaving some intrigue or other, keen in the pursuit of wisdom, fertile in resources; a philosopher at all times, terrible as an enchanter, sorcerer, sophist. He is by nature neither mortal nor immortal, but alive and flourishing at one moment when he is in plenty, and dead at another moment, and again alive by reason of his father's nature. But that which is always flowing in is always flowing out, and so he is never in want and never in wealth; and, further, he is in a mean between ignorance and knowledge. The truth of the matter is this: No god is a philosopher or seeker after wisdom, for he is wise already; nor does any man who is wise seek after wisdom.  For herein is the evil of ignorance, that he who is neither good nor wise is nevertheless satisfied with himself: he has no desire for that of which he feels no want.' 'But who then, Diotima,' I said, 'are the lovers of wisdom, if they are neither the wise nor the foolish?' 'A child may answer that question,' she replied; 'they are those who are in a mean between the two; Love is one of them. For wisdom is a most beautiful thing, and Love is of the beautiful; and therefore Love is also a philosopher or lover of wisdom, and being a lover of wisdom is in a mean between the wise and the ignorant. And of this too his birth is the cause; for his father is wealthy and wise, and his mother poor and foolish. Such, my dear Socrates, is the nature of the spirit Love. The error in your conception of him was very natural, and as I imagine from what you say, has arisen out of a confusion of love and the beloved, which made you think that love was all beautiful. For the beloved is the truly beautiful, and delicate, and perfect, and blessed; but the principle of love is of another nature, and is such as I have described.'
I said, 'O thou stranger woman, thou sayest well; but, assuming Love to be such as you say, what is the use of him to men?' 'That, Socrates,' she replied, 'I will attempt to unfold: of his nature and birth I have already spoken; and you acknowledge that love is of the beautiful. But some one will say: Of the beautiful in what, Socrates and Diotima?—or rather let me put the question more clearly, and ask: When a man loves the beautiful, what does he desire?' I answered her 'That the beautiful may be his.' 'Still,' she said, 'the answer suggests a further question: What is given by the possession of beauty?' 'To what you have asked,' I replied, 'I have no answer ready.' 'Then,' she said, 'let me put the word "good" in the place of the beautiful, and repeat the question once more: If he who loves loves the good, what is it then that he loves?' 'The possession of the good,' I said. 'And what does he gain who possesses the good?' 'Happiness,' I replied; 'there is less difficulty in answering that question.' 'Yes,' she said, 'the happy are made happy by the acquisition of good things. Nor is there any need to ask why a man desires happiness; the answer is already final.' 'You are right.' I said. 'And is this wish and this desire common to all? and do all men always desire their own good, or only some men?—what say you?' 'All men,' I replied; 'the desire is common to all.' 'Why, then,' she rejoined, 'are not all men, Socrates, said to love, but only some of them? whereas you say that all men are always loving the same things.' 'I myself wonder,' I said, 'why this is.' 'There is nothing to wonder at,' she replied; 'the reason is that one part of love is separated off and receives the name of the whole, but the other parts have other names.' 'Give an illustration,' I said. She answered me as follows: 'There is poetry, which, as you know, is complex and manifold. All creation or passage of non-being into being is poetry or making, and the processes of all art are creative; and the masters of arts are all poets or makers.' 'Very true.' 'Still,' she said, 'you know that they are not called poets, but have other names; only that portion of the art which is separated off from the rest, and is concerned with music and metre, is termed poetry, and they who possess poetry in this sense of the word are called poets.' 'Very true,' I said. 'And the same holds of love. For you may say generally that all desire of good and happiness is only the great and subtle power of love; but they who are drawn towards him by any other path, whether the path of money-making or gymnastics or philosophy, are not called lovers—the name of the whole is appropriated to those whose affection takes one form only—they alone are said to love, or to be lovers.' 'I dare say,' I replied, 'that you are right.' 'Yes,' she added, 'and you hear people say that lovers are seeking for their other half; but I say that they are seeking neither for the half of themselves, nor for the whole, unless the half or the whole be also a good. And they will cut off their own hands and feet and cast them away, if they are evil; for they love not what is their own, unless perchance there be some one who calls what belongs to him the good, and what belongs to another the evil. For there is nothing which men love but the good. Is there anything?' 'Certainly, I should say, that there is nothing.' 'Then,' she said, 'the simple truth is, that men love the good.' 'Yes,' I said. 'To which must be added that they love the possession of the good?' 'Yes, that must be added.' 'And not only the possession, but the everlasting possession of the good?' 'That must be added too.' 'Then love,' she said, 'may be described generally as the love of the everlasting possession of the good?' 'That is most true.'
'Then if this be the nature of love, can you tell me further,' she said, 'what is the manner of the pursuit? what are they doing who show all this eagerness and heat which is called love? and what is the object which they have in view? Answer me.' 'Nay, Diotima,' I replied, 'if I had known, I should not have wondered at your wisdom, neither should I have come to learn from you about this very matter.' 'Well,' she said, 'I will teach you:—The object which they have in view is birth in beauty, whether of body or soul.' 'I do not understand you,' I said; 'the oracle requires an explanation.' 'I will make my meaning clearer,' she replied. 'I mean to say, that all men are bringing to the birth in their bodies and in their souls. There is a certain age at which human nature is desirous of procreation—procreation which must be in beauty and not in deformity; and this procreation is the union of man and woman, and is a divine thing; for conception and generation are an immortal principle in the mortal creature, and in the inharmonious they can never be. But the deformed is always inharmonious with the divine, and the beautiful harmonious. Beauty, then, is the destiny or goddess of parturition who presides at birth, and therefore, when approaching beauty, the conceiving power is propitious, and diffusive, and benign, and begets and bears fruit: at the sight of ugliness she frowns and contracts and has a sense of pain, and turns away, and shrivels up, and not without a pang refrains from conception. And this is the reason why, when the hour of conception arrives, and the teeming nature is full, there is such a flutter and ecstasy about beauty whose approach is the alleviation of the pain of travail. For love, Socrates, is not, as you imagine, the love of the beautiful only.' 'What then?' 'The love of generation and of birth in beauty.' 'Yes,' I said. 'Yes, indeed,' she replied. 'But why of generation?' 'Because to the mortal creature, generation is a sort of eternity and immortality,' she replied; 'and if, as has been already admitted, love is of the everlasting possession of the good, all men will necessarily desire immortality together with good: Wherefore love is of immortality.'
All this she taught me at various times when she spoke of love. And I remember her once saying to me, 'What is the cause, Socrates, of love, and the attendant desire? See you not how all animals, birds, as well as beasts, in their desire of procreation, are in agony when they take the infection of love, which begins with the desire of union; whereto is added the care of offspring, on whose behalf the weakest are ready to battle against the strongest even to the uttermost, and to die for them, and will let themselves be tormented with hunger or suffer anything in order to maintain their young. Man may be supposed to act thus from reason; but why should animals have these passionate feelings? Can you tell me why?' Again I replied that I did not know. She said to me: 'And do you expect ever to become a master in the art of love, if you do not know this?' 'But I have told you already, Diotima, that my ignorance is the reason why I come to you; for I am conscious that I want a teacher; tell me then the cause of this and of the other mysteries of love.' 'Marvel not,' she said, 'if you believe that love is of the immortal, as we have several times acknowledged; for here again, and on the same principle too, the mortal nature is seeking as far as is possible to be everlasting and immortal: and this is only to be attained by generation, because generation always leaves behind a new existence in the place of the old. Nay even in the life of the same individual there is succession and not absolute unity: a man is called the same, and yet in the short interval which elapses between youth and age, and in which every animal is said to have life and identity, he is undergoing a perpetual process of loss and reparation—hair, flesh, bones, blood, and the whole body are always changing. Which is true not only of the body, but also of the soul, whose habits, tempers, opinions, desires, pleasures, pains, fears, never remain the same in any one of us, but are always coming and going; and equally true of knowledge, and what is still more surprising to us mortals, not only do the sciences in general spring up and decay, so that in respect of them we are never the same; but each of them individually experiences a like change. For what is implied in the word "recollection," but the departure of knowledge, which is ever being forgotten, and is renewed and preserved by recollection, and appears to be the same although in reality new, according to that law of succession by which all mortal things are preserved, not absolutely the same, but by substitution, the old worn-out mortality leaving another new and similar existence behind—unlike the divine, which is always the same and not another? And in this way, Socrates, the mortal body, or mortal anything, partakes of immortality; but the immortal in another way. Marvel not then at the love which all men have of their offspring; for that universal love and interest is for the sake of immortality.'
I was astonished at her words, and said: 'Is this really true, O thou wise Diotima?' And she answered with all the authority of an accomplished sophist: 'Of that, Socrates, you may be assured;—think only of the ambition of men, and you will wonder at the senselessness of their ways, unless you consider how they are stirred by the love of an immortality of fame.  Do you imagine that Alcestis would have died to save Admetus, or Achilles to avenge Patroclus, or your own Codrus in order to preserve the kingdom for his sons, if they had not imagined that the memory of their virtues, which still survives among us, would be immortal? Nay,' she said, 'I am persuaded that all men do all things, and the better they are the more they do them, in hope of the glorious fame of immortal virtue; for they desire the immortal.
'Those who are pregnant in the body only, betake themselves to women and beget children—this is the character of their love; their offspring, as they hope, will preserve their memory and giving them the blessedness and immortality which they desire in the future. But souls which are pregnant—for there certainly are men who are more creative in their souls than in their bodies—conceive that which is proper for the soul to conceive or contain. And what are these conceptions?—wisdom and virtue in general. And such creators are poets and all artists who are deserving of the name inventor. But the greatest and fairest sort of wisdom by far is that which is concerned with the ordering of states and families, and which is called temperance and justice. And he who in youth has the seed of these implanted in him and is himself inspired, when he comes to maturity desires to beget and generate. He wanders about seeking beauty that he may beget offspring—for in deformity he will beget nothing—and naturally embraces the beautiful rather than the deformed body; above all when he finds a fair and noble and well-nurtured soul, he embraces the two in one person, and to such an one he is full of speech about virtue and the nature and pursuits of a good man; and he tries to educate him; and at the touch of the beautiful which is ever present to his memory, even when absent, he brings forth that which he had conceived long before, and in company with him tends that which he brings forth; and they are married by a far nearer tie and have a closer friendship than those who beget mortal children, for the children who are their common offspring are fairer and more immortal. Who, when he thinks of Homer and Hesiod and other great poets, would not rather have their children than ordinary human ones? Who would not emulate them in the creation of children such as theirs, which have preserved their memory and given them everlasting glory? Or who would not have such children as Lycurgus left behind him to be the saviours, not only of Lacedaemon, but of Hellas, as one may say? There is Solon, too, who is the revered father of Athenian laws; and many others there are in many other places, both among Hellenes and barbarians, who have given to the world many noble works, and have been the parents of virtue of every kind; and many temples have been raised in their honour for the sake of children such as theirs; which were never raised in honour of any one, for the sake of his mortal children.
'These are the lesser mysteries of love, into which even you, Socrates, may enter; to the greater and more hidden ones which are the crown of these, and to which, if you pursue them in a right spirit, they will lead, I know not whether you will be able to attain. But I will do my utmost to inform you, and do you follow if you can. For he who would proceed aright in this matter should begin in youth to visit beautiful forms; and first, if he be guided by his instructor aright, to love one such form only—out of that he should create fair thoughts; and soon he will of himself perceive that the beauty of one form is akin to the beauty of another; and then if beauty of form in general is his pursuit, how foolish would he be not to recognize that the beauty in every form is and the same! And when he perceives this he will abate his violent love of the one, which he will despise and deem a small thing, and will become a lover of all beautiful forms; in the next stage he will consider that the beauty of the mind is more honourable than the beauty of the outward form. So that if a virtuous soul have but a little comeliness, he will be content to love and tend him, and will search out and bring to the birth thoughts which may improve the young, until he is compelled to contemplate and see the beauty of institutions and laws, and to understand that the beauty of them all is of one family, and that personal beauty is a trifle; and after laws and institutions he will go on to the sciences, that he may see their beauty, being not like a servant in love with the beauty of one youth or man or institution, himself a slave mean and narrow-minded, but drawing towards and contemplating the vast sea of beauty, he will create many fair and noble thoughts and notions in boundless love of wisdom; until on that shore he grows and waxes strong, and at last the vision is revealed to him of a single science, which is the science of beauty everywhere. To this I will proceed; please to give me your very best attention:
'He who has been instructed thus far in the things of love, and who has learned to see the beautiful in due order and succession, when he comes toward the end will suddenly perceive a nature of wondrous beauty (and this, Socrates, is the final cause of all our former toils)—a nature which in the first place is everlasting, not growing and decaying, or waxing and waning; secondly, not fair in one point of view and foul in another, or at one time or in one relation or at one place fair, at another time or in another relation or at another place foul, as if fair to some and foul to others, or in the likeness of a face or hands or any other part of the bodily frame, or in any form of speech or knowledge, or existing in any other being, as for example, in an animal, or in heaven, or in earth, or in any other place; but beauty absolute, separate, simple, and everlasting, which without diminution and without increase, or any change, is imparted to the ever-growing and perishing beauties of all other things. He who from these ascending under the influence of true love, begins to perceive that beauty, is not far from the end. And the true order of going, or being led by another, to the things of love, is to begin from the beauties of earth and mount upwards for the sake of that other beauty, using these as steps only, and from one going on to two, and from two to all fair forms, and from fair forms to fair practices, and from fair practices to fair notions, until from fair notions he arrives at the notion of absolute beauty, and at last knows what the essence of beauty is. This, my dear Socrates,' said the stranger of Mantineia, 'is that life above all others which man should live, in the contemplation of beauty absolute; a beauty which if you once beheld, you would see not to be after the measure of gold, and garments, and fair boys and youths, whose presence now entrances you; and you and many a one would be content to live seeing them only and conversing with them without meat or drink, if that were possible—you only want to look at them and to be with them. But what if man had eyes to see the true beauty—the divine beauty, I mean, pure and clear and unalloyed, not clogged with the pollutions of mortality and all the colours and vanities of human life—thither looking, and holding converse with the true beauty simple and divine? Remember how in that communion only, beholding beauty with the eye of the mind, he will be enabled to bring forth, not images of beauty, but realities (for he has hold not of an image but of a reality), and bringing forth and nourishing true virtue to become the friend of God and be immortal, if mortal man may. Would that be an ignoble life?'
Such, Phaedrus—and I speak not only to you, but to all of you—were the words of Diotima; and I am persuaded of their truth. And being persuaded of them, I try to persuade others, that in the attainment of this end human nature will not easily find a helper better than love: And therefore, also, I say that every man ought to honour him as I myself honour him, and walk in his ways, and exhort others to do the same, and praise the power and spirit of love according to the measure of my ability now and ever.
The words which I have spoken, you, Phaedrus, may call an encomium of love, or anything else which you please.
When Socrates had done speaking, the company applauded, and Aristophanes was beginning to say something in answer to the allusion which Socrates had made to his own speech, when suddenly there was a great knocking at the door of the house, as of revellers, and the sound of a flute-girl was heard. Agathon told the attendants to go and see who were the intruders. 'If they are friends of ours,' he said, 'invite them in, but if not, say that the drinking is over.' A little while afterwards they heard the voice of Alcibiades resounding in the court; he was in a great state of intoxication, and kept roaring and shouting 'Where is Agathon? Lead me to Agathon,' and at length, supported by the flute-girl and some of his attendants, he found his way to them. 'Hail, friends,' he said, appearing at the door crowned with a massive garland of ivy and violets, his head flowing with ribands. 'Will you have a very drunken man as a companion of your revels? Or shall I crown Agathon, which was my intention in coming, and go away? For I was unable to come yesterday, and therefore I am here to-day, carrying on my head these ribands, that taking them from my own head, I may crown the head of this fairest and wisest of men, as I may be allowed to call him.  Yet I know very well that I am speaking the truth, although you may laugh. But first tell me; if I come in shall we have the understanding of which I spoke (supra Will you have a very drunken man? etc.)? Will you drink with me or not?'
The company were vociferous in begging that he would take his place among them, and Agathon specially invited him. Thereupon he was led in by the people who were with him; and as he was being led, intending to crown Agathon, he took the ribands from his own head and held them in front of his eyes; he was thus prevented from seeing Socrates, who made way for him, and Alcibiades took the vacant place between Agathon and Socrates, and in taking the place he embraced Agathon and crowned him. Take off his sandals, said Agathon, and let him make a third on the same couch.
By all means; but who makes the third partner in our revels? said Alcibiades, turning round and starting up as he caught sight of Socrates. By Heracles, he said, what is this? here is Socrates always lying in wait for me, and always, as his way is, coming out at all sorts of unsuspected places: and now, what have you to say for yourself, and why are you lying here, where I perceive that you have contrived to find a place, not by a joker or lover of jokes, like Aristophanes, but by the fairest of the company?
Socrates turned to Agathon and said: I must ask you to protect me, Agathon; for the passion of this man has grown quite a serious matter to me. Since I became his admirer I have never been allowed to speak to any other fair one, or so much as to look at them. If I do, he goes wild with envy and jealousy, and not only abuses me but can hardly keep his hands off me, and at this moment he may do me some harm. Please to see to this, and either reconcile me to him, or, if he attempts violence, protect me, as I am in bodily fear of his mad and passionate attempts.
There can never be reconciliation between you and me, said Alcibiades; but for the present I will defer your chastisement. And I must beg you, Agathon, to give me back some of the ribands that I may crown the marvellous head of this universal despot—I would not have him complain of me for crowning you, and neglecting him, who in conversation is the conqueror of all mankind; and this not only once, as you were the day before yesterday, but always. Whereupon, taking some of the ribands, he crowned Socrates, and again reclined.
Then he said: You seem, my friends, to be sober, which is a thing not to be endured; you must drink—for that was the agreement under which I was admitted—and I elect myself master of the feast until you are well drunk. Let us have a large goblet, Agathon, or rather, he said, addressing the attendant, bring me that wine-cooler. The wine-cooler which had caught his eye was a vessel holding more than two quarts—this he filled and emptied, and bade the attendant fill it again for Socrates. Observe, my friends, said Alcibiades, that this ingenious trick of mine will have no effect on Socrates, for he can drink any quantity of wine and not be at all nearer being drunk. Socrates drank the cup which the attendant filled for him.
Eryximachus said: What is this, Alcibiades? Are we to have neither conversation nor singing over our cups; but simply to drink as if we were thirsty?
Alcibiades replied: Hail, worthy son of a most wise and worthy sire!
The same to you, said Eryximachus; but what shall we do?
That I leave to you, said Alcibiades.
'The wise physician skilled our wounds to heal (from Pope's Homer, Il.)'
shall prescribe and we will obey. What do you want?
Well, said Eryximachus, before you appeared we had passed a resolution that each one of us in turn should make a speech in praise of love, and as good a one as he could: the turn was passed round from left to right; and as all of us have spoken, and you have not spoken but have well drunken, you ought to speak, and then impose upon Socrates any task which you please, and he on his right hand neighbour, and so on.
That is good, Eryximachus, said Alcibiades; and yet the comparison of a drunken man's speech with those of sober men is hardly fair; and I should like to know, sweet friend, whether you really believe what Socrates was just now saying; for I can assure you that the very reverse is the fact, and that if I praise any one but himself in his presence, whether God or man, he will hardly keep his hands off me.
For shame, said Socrates.
Hold your tongue, said Alcibiades, for by Poseidon, there is no one else whom I will praise when you are of the company.
Well then, said Eryximachus, if you like praise Socrates.
What do you think, Eryximachus? said Alcibiades: shall I attack him and inflict the punishment before you all?
What are you about? said Socrates; are you going to raise a laugh at my expense? Is that the meaning of your praise?
I am going to speak the truth, if you will permit me.
I not only permit, but exhort you to speak the truth.
Then I will begin at once, said Alcibiades, and if I say anything which is not true, you may interrupt me if you will, and say 'that is a lie,' though my intention is to speak the truth. But you must not wonder if I speak any how as things come into my mind; for the fluent and orderly enumeration of all your singularities is not a task which is easy to a man in my condition.
And now, my boys, I shall praise Socrates in a figure which will appear to him to be a caricature, and yet I speak, not to make fun of him, but only for the truth's sake. I say, that he is exactly like the busts of Silenus, which are set up in the statuaries' shops, holding pipes and flutes in their mouths; and they are made to open in the middle, and have images of gods inside them. I say also that he is like Marsyas the satyr.  Aye, and there is a resemblance in other points too. For example, you are a bully, as I can prove by witnesses, if you will not confess. And are you not a flute-player? That you are, and a performer far more wonderful than Marsyas. He indeed with instruments used to charm the souls of men by the power of his breath, and the players of his music do so still: for the melodies of Olympus (compare Arist. Pol.) are derived from Marsyas who taught them, and these, whether they are played by a great master or by a miserable flute-girl, have a power which no others have; they alone possess the soul and reveal the wants of those who have need of gods and mysteries, because they are divine. But you produce the same effect with your words only, and do not require the flute: that is the difference between you and him. When we hear any other speaker, even a very good one, he produces absolutely no effect upon us, or not much, whereas the mere fragments of you and your words, even at second-hand, and however imperfectly repeated, amaze and possess the souls of every man, woman, and child who comes within hearing of them. And if I were not afraid that you would think me hopelessly drunk, I would have sworn as well as spoken to the influence which they have always had and still have over me. For my heart leaps within me more than that of any Corybantian reveller, and my eyes rain tears when I hear them. And I observe that many others are affected in the same manner. I have heard Pericles and other great orators, and I thought that they spoke well, but I never had any similar feeling; my soul was not stirred by them, nor was I angry at the thought of my own slavish state. But this Marsyas has often brought me to such a pass, that I have felt as if I could hardly endure the life which I am leading (this, Socrates, you will admit); and I am conscious that if I did not shut my ears against him, and fly as from the voice of the siren, my fate would be like that of others,—he would transfix me, and I should grow old sitting at his feet. For he makes me confess that I ought not to live as I do, neglecting the wants of my own soul, and busying myself with the concerns of the Athenians; therefore I hold my ears and tear myself away from him. And he is the only person who ever made me ashamed, which you might think not to be in my nature, and there is no one else who does the same. For I know that I cannot answer him or say that I ought not to do as he bids, but when I leave his presence the love of popularity gets the better of me. And therefore I run away and fly from him, and when I see him I am ashamed of what I have confessed to him. Many a time have I wished that he were dead, and yet I know that I should be much more sorry than glad, if he were to die: so that I am at my wit's end.
And this is what I and many others have suffered from the flute-playing of this satyr. Yet hear me once more while I show you how exact the image is, and how marvellous his power. For let me tell you; none of you know him; but I will reveal him to you; having begun, I must go on. See you how fond he is of the fair?  Is he not like a Silenus in this? To be sure he is: his outer mask is the carved head of the Silenus; but, O my companions in drink, when he is opened, what temperance there is residing within! Know you that beauty and wealth and honour, at which the many wonder, are of no account with him, and are utterly despised by him: he regards not at all the persons who are gifted with them; mankind are nothing to him; all his life is spent in mocking and flouting at them. But when I opened him, and looked within at his serious purpose, I saw in him divine and golden images of such fascinating beauty that I was ready to do in a moment whatever Socrates commanded: they may have escaped the observation of others, but I saw them. Now I fancied that he was seriously enamoured of my beauty, and I thought that I should therefore have a grand opportunity of hearing him tell what he knew, for I had a wonderful opinion of the attractions of my youth. In the prosecution of this design, when I next went to him, I sent away the attendant who usually accompanied me (I will confess the whole truth, and beg you to listen; and if I speak falsely, do you, Socrates, expose the falsehood). Well, he and I were alone together, and I thought that when there was nobody with us, I should hear him speak the language which lovers use to their loves when they are by themselves, and I was delighted. Nothing of the sort; he conversed as usual, and spent the day with me and then went away. Afterwards I challenged him to the palaestra; and he wrestled and closed with me several times when there was no one present; I fancied that I might succeed in this manner. Not a bit; I made no way with him. Lastly, as I had failed hitherto, I thought that I must take stronger measures and attack him boldly, and, as I had begun, not give him up, but see how matters stood between him and me. So I invited him to sup with me, just as if he were a fair youth, and I a designing lover. He was not easily persuaded to come; he did, however, after a while accept the invitation, and when he came the first time, he wanted to go away at once as soon as supper was over, and I had not the face to detain him. The second time, still in pursuance of my design, after we had supped, I went on conversing far into the night, and when he wanted to go away, I pretended that the hour was late and that he had much better remain. So he lay down on the couch next to me, the same on which he had supped, and there was no one but ourselves sleeping in the apartment. All this may be told without shame to any one. But what follows I could hardly tell you if I were sober. Yet as the proverb says, 'In vino veritas,' whether with boys, or without them (In allusion to two proverbs.); and therefore I must speak. Nor, again, should I be justified in concealing the lofty actions of Socrates when I come to praise him. Moreover I have felt the serpent's sting; and he who has suffered, as they say, is willing to tell his fellow-sufferers only, as they alone will be likely to understand him, and will not be extreme in judging of the sayings or doings which have been wrung from his agony. For I have been bitten by a more than viper's tooth; I have known in my soul, or in my heart, or in some other part, that worst of pangs, more violent in ingenuous youth than any serpent's tooth, the pang of philosophy, which will make a man say or do anything. And you whom I see around me, Phaedrus and Agathon and Eryximachus and Pausanias and Aristodemus and Aristophanes, all of you, and I need not say Socrates himself, have had experience of the same madness and passion in your longing after wisdom. Therefore listen and excuse my doings then and my sayings now. But let the attendants and other profane and unmannered persons close up the doors of their ears.
When the lamp was put out and the servants had gone away, I thought that I must be plain with him and have no more ambiguity.  'Do you know what I am meditating? 'What are you meditating?' he said.  Now I feel that I should be a fool to refuse you this or any other favour, and therefore I come to lay at your feet all that I have and all that my friends have, in the hope that you will assist me in the way of virtue, which I desire above all things, and in which I believe that you can help me better than any one else. And I should certainly have more reason to be ashamed of what wise men would say if I were to refuse a favour to such as you, than of what the world, who are mostly fools, would say of me if I granted it.' To these words he replied in the ironical manner which is so characteristic of him:—'Alcibiades, my friend, you have indeed an elevated aim if what you say is true, and if there really is in me any power by which you may become better; truly you must see in me some rare beauty of a kind infinitely higher than any which I see in you. And therefore, if you mean to share with me and to exchange beauty for beauty, you will have greatly the advantage of me; you will gain true beauty in return for appearance—like Diomede, gold in exchange for brass. But look again, sweet friend, and see whether you are not deceived in me. The mind begins to grow critical when the bodily eye fails, and it will be a long time before you get old.' Hearing this, I said: 'I have told you my purpose, which is quite serious, and do you consider what you think best for you and me.' 'That is good,' he said; 'at some other time then we will consider and act as seems best about this and about other matters.' Whereupon, I fancied that he was smitten, and that the words which I had uttered like arrows had wounded him, and so without waiting to hear more I got up, and throwing my coat about him crept under his threadbare cloak, as the time of year was winter, and there I lay during the whole night having this wonderful monster in my arms. This again, Socrates, will not be denied by you. And yet, notwithstanding all, he was so superior to my solicitations, so contemptuous and derisive and disdainful of my beauty—which really, as I fancied, had some attractions—hear, O judges; for judges you shall be of the haughty virtue of Socrates—nothing more happened, but in the morning when I awoke (let all the gods and goddesses be my witnesses) I arose as from the couch of a father or an elder brother.
What do you suppose must have been my feelings, after this rejection, at the thought of my own dishonour? And yet I could not help wondering at his natural temperance and self-restraint and manliness. I never imagined that I could have met with a man such as he is in wisdom and endurance. And therefore I could not be angry with him or renounce his company, any more than I could hope to win him. For I well knew that if Ajax could not be wounded by steel, much less he by money; and my only chance of captivating him by my personal attractions had failed. So I was at my wit's end; no one was ever more hopelessly enslaved by another. All this happened before he and I went on the expedition to Potidaea; there we messed together, and I had the opportunity of observing his extraordinary power of sustaining fatigue. His endurance was simply marvellous when, being cut off from our supplies, we were compelled to go without food—on such occasions, which often happen in time of war, he was superior not only to me but to everybody; there was no one to be compared to him. Yet at a festival he was the only person who had any real powers of enjoyment; though not willing to drink, he could if compelled beat us all at that,—wonderful to relate! no human being had ever seen Socrates drunk; and his powers, if I am not mistaken, will be tested before long. His fortitude in enduring cold was also surprising. There was a severe frost, for the winter in that region is really tremendous, and everybody else either remained indoors, or if they went out had on an amazing quantity of clothes, and were well shod, and had their feet swathed in felt and fleeces: in the midst of this, Socrates with his bare feet on the ice and in his ordinary dress marched better than the other soldiers who had shoes, and they looked daggers at him because he seemed to despise them.
I have told you one tale, and now I must tell you another, which is worth hearing,
'Of the doings and sufferings of the enduring man'
while he was on the expedition. One morning he was thinking about something which he could not resolve; he would not give it up, but continued thinking from early dawn until noon—there he stood fixed in thought; and at noon attention was drawn to him, and the rumour ran through the wondering crowd that Socrates had been standing and thinking about something ever since the break of day. At last, in the evening after supper, some Ionians out of curiosity (I should explain that this was not in winter but in summer), brought out their mats and slept in the open air that they might watch him and see whether he would stand all night. There he stood until the following morning; and with the return of light he offered up a prayer to the sun, and went his way (compare supra). I will also tell, if you please—and indeed I am bound to tell—of his courage in battle; for who but he saved my life? Now this was the engagement in which I received the prize of valour: for I was wounded and he would not leave me, but he rescued me and my arms; and he ought to have received the prize of valour which the generals wanted to confer on me partly on account of my rank, and I told them so, (this, again, Socrates will not impeach or deny), but he was more eager than the generals that I and not he should have the prize. There was another occasion on which his behaviour was very remarkable—in the flight of the army after the battle of Delium, where he served among the heavy-armed,—I had a better opportunity of seeing him than at Potidaea, for I was myself on horseback, and therefore comparatively out of danger. He and Laches were retreating, for the troops were in flight, and I met them and told them not to be discouraged, and promised to remain with them; and there you might see him, Aristophanes, as you describe (Aristoph. Clouds), just as he is in the streets of Athens, stalking like a pelican, and rolling his eyes, calmly contemplating enemies as well as friends, and making very intelligible to anybody, even from a distance, that whoever attacked him would be likely to meet with a stout resistance; and in this way he and his companion escaped—for this is the sort of man who is never touched in war; those only are pursued who are running away headlong. I particularly observed how superior he was to Laches in presence of mind. Many are the marvels which I might narrate in praise of Socrates; most of his ways might perhaps be paralleled in another man, but his absolute unlikeness to any human being that is or ever has been is perfectly astonishing. You may imagine Brasidas and others to have been like Achilles; or you may imagine Nestor and Antenor to have been like Pericles; and the same may be said of other famous men, but of this strange being you will never be able to find any likeness, however remote, either among men who now are or who ever have been—other than that which I have already suggested of Silenus and the satyrs; and they represent in a figure not only himself, but his words. For, although I forgot to mention this to you before, his words are like the images of Silenus which open; they are ridiculous when you first hear them; he clothes himself in language that is like the skin of the wanton satyr—for his talk is of pack-asses and smiths and cobblers and curriers, and he is always repeating the same things in the same words (compare Gorg.), so that any ignorant or inexperienced person might feel disposed to laugh at him; but he who opens the bust and sees what is within will find that they are the only words which have a meaning in them, and also the most divine, abounding in fair images of virtue, and of the widest comprehension, or rather extending to the whole duty of a good and honourable man.
This, friends, is my praise of Socrates. I have added my blame of him for his ill-treatment of me; and he has ill-treated not only me, but Charmides the son of Glaucon, and Euthydemus the son of Diocles, and many others in the same way—beginning as their lover he has ended by making them pay their addresses to him. Wherefore I say to you, Agathon, 'Be not deceived by him; learn from me and take warning, and do not be a fool and learn by experience, as the proverb says.'
When Alcibiades had finished, there was a laugh at his outspokenness; for he seemed to be still in love with Socrates. You are sober, Alcibiades, said Socrates, or you would never have gone so far about to hide the purpose of your satyr's praises, for all this long story is only an ingenious circumlocution, of which the point comes in by the way at the end; you want to get up a quarrel between me and Agathon, and your notion is that I ought to love you and nobody else, and that you and you only ought to love Agathon. But the plot of this Satyric or Silenic drama has been detected, and you must not allow him, Agathon, to set us at variance.
I believe you are right, said Agathon, and I am disposed to think that his intention in placing himself between you and me was only to divide us; but he shall gain nothing by that move; for I will go and lie on the couch next to you.
Yes, yes, replied Socrates, by all means come here and lie on the couch below me.
Alas, said Alcibiades, how I am fooled by this man; he is determined to get the better of me at every turn. I do beseech you, allow Agathon to lie between us.
Certainly not, said Socrates, as you praised me, and I in turn ought to praise my neighbour on the right, he will be out of order in praising me again when he ought rather to be praised by me, and I must entreat you to consent to this, and not be jealous, for I have a great desire to praise the youth.
Hurrah! cried Agathon, I will rise instantly, that I may be praised by Socrates.
The usual way, said Alcibiades; where Socrates is, no one else has any chance with the fair; and now how readily has he invented a specious reason for attracting Agathon to himself.
Agathon arose in order that he might take his place on the couch by Socrates, when suddenly a band of revellers entered, and spoiled the order of the banquet. Some one who was going out having left the door open, they had found their way in, and made themselves at home; great confusion ensued, and every one was compelled to drink large quantities of wine. Aristodemus said that Eryximachus, Phaedrus, and others went away—he himself fell asleep, and as the nights were long took a good rest: he was awakened towards daybreak by a crowing of cocks, and when he awoke, the others were either asleep, or had gone away; there remained only Socrates, Aristophanes, and Agathon, who were drinking out of a large goblet which they passed round, and Socrates was discoursing to them. Aristodemus was only half awake, and he did not hear the beginning of the discourse; the chief thing which he remembered was Socrates compelling the other two to acknowledge that the genius of comedy was the same with that of tragedy, and that the true artist in tragedy was an artist in comedy also. To this they were constrained to assent, being drowsy, and not quite following the argument. And first of all Aristophanes dropped off, then, when the day was already dawning, Agathon. Socrates, having laid them to sleep, rose to depart; Aristodemus, as his manner was, following him. At the Lyceum he took a bath, and passed the day as usual. In the evening he retired to rest at his own home.


9.23: Putting It Together- Communicating through Technology
Now that you’ve learned the basics about communicating through technology today, you must remember that it’s always changing. A year from now there will be platforms and apps available that haven’t even been thought of yet. Your main goal is to remember these basic ideas no matter what technology you use the following will remain the same:
- The point of the meeting is the content of the discussion or presentation, not the technology itself.
- Your colleagues’ comfort level with technology will range pretty widely. Be patient with them.
- The app that’s scaring or frustrating you today will eventually be as easy as email. Be patient with yourself.
A successful interaction using any of the technologies covered in this module meets a few basic criteria, many of which are designed to alleviate the kinds of bad surprises technology can sometimes involve.
In general, to ensure a successful interaction using communication technology, the host or planner has done the following:
- Practiced with the technology.
- Planned the scheduling of the meeting to include enough time for participants to download software, if necessary, and make sure it runs on their systems.
- Sent clear instructions to participants well in advance and done her best to simplify the process for them as much as possible.
- Created at least one back-up plan—preferably two—in case the technology doesn’t work for everyone.
- Planned the timing of the meeting to allow not only for the content to be covered but also for the additional minutes needed to be sure everyone is up and running.
- Sent an agenda to participants that includes time for the technology.
- Learned how to present information, conduct discussion, and generate next steps appropriately and effectively within the technology platform being used.
- Conceived a plan to follow up with information, notes, conclusions, and next steps after the meeting has concluded.
The point of communication through technology is to get work done efficiently, and the best way to do that is to have a toolbox of possibilities that you feel comfortable with and that meet your needs. The ability to communicate through such a wide variety of channels opens up your work life and helps you be a better leader, employee, and colleague.
Contributors and Attributions
- Putting It Together: Communicating through Technology. Authored by : Barbara Egel. Provided by : Lumen Learning. License : CC BY: Attribution


18.8: The Future of Psychology- Bandura's Vision as Compared to Freud
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In Toward a Psychology of Human Agency, Bandura (2006) agrees with Freud that religion played a significant role in the advancement of civilization, and that the scientific revolution that began with Darwin’s Theory of Evolution shifted our focus toward natural and scientific approaches to understanding and improving human life. However, evolution emphasizes random effects on genetically determined structures and traits, and does not allow for the choices made by individuals, though sociobiologists argue that genetics can exert influences on those choices. Rather than relying on psychoanalysis in order to understand the impulses of the id that drive us to survive, as Freud proposed (1927/1961, 1930/1961), Bandura believes that people are agents of change in their own lives, and that they can choose the direction that change takes.
Being an agent involves intentionally influencing one’s functioning and life circumstances, and there are four core properties of human agency. Intentionality refers to our ability to form action plans and the strategies necessary for accomplishing them. Forethought is the temporal extension of agency, in which we set goals for ourselves and anticipate likely outcomes of our actions to both guide and motivate our behavior. Once having chosen a course of action, of course, we do not simply sit back and wait for the right things to happen. Agency also involves both self-reactiveness and self-reflectiveness , processes in which we regulate our behavior, monitor our courses of action, and examine whether we are capable of being successful in our various endeavors (Bandura, 2006). As much as human agency involves our own thoughts, goals, motivation, and expectations, however, we do not exist autonomously. All human behavior occurs within social structures, and there is a reciprocal interplay between personal, behavioral, and environmental determinants (reciprocal determinism). This means that human agency, the exercise of self-influence, is part of the causal structure of our lives. As Bandura points out, this is not “free will,” which would be a throwback to medieval theology, but rather a matter of acting as an agent, the role of an individual in making causal contributions to the course of events in their life (Bandura, 2006).
…Social cognitive theory rejects a duality of human agency and a disembodied social structure. Social systems are the product of human activity, and social systems, in turn, help to organize, guide, and regulate human affairs. However, in the dynamic interplay within the societal rule structures, there is considerable personal variation in the interpretation of, adoption of, enforcement of, circumvention of, and opposition to societal prescriptions and sanctions…freedom is conceived not just passively as the absence of constraints, but also proactively as the exercise of self-influence… (pg. 165; Bandura, 2006)


15.1: Projective Completion
In the Euclidean plane, two distinct lines might have one or zero points of intersection (in the latter case the lines are parallel). Our aim is to extend the Euclidean plane by ideal points so that any two distinct lines will have exactly one point of intersection.
A collection of lines in the Euclidean plane is called concurrent if they all intersect at a single point or all of them pairwise parallel. A maximal set of concurrent lines in the plane is called pencil . There are two types of pencils: central pencils contain all lines passing thru a fixed point called the center of the pencil and parallel pencil contain pairwise parallel lines.
Each point in the Euclidean plane uniquely defines a central pencil with the center in it. Note that any two lines completely determine the pencil containing both.
Let us add one ideal point for each parallel pencil, and assume that all these ideal points lie on one ideal line . We also assume that the ideal line belongs to each parallel pencil.
We obtain the so-called real projective plane , or projective completion of the original plane. It comes with an incidence structure — we say that three points lie on one line if the corresponding pencils contain a common line. Projective geometry studies this incidence structure.
Let us describe points of projective completion in coordinates. A parallel pencil contains the ideal line and the lines \(y = m \cdot x+b\) with fixed slope \(m\) ; if \(m=\infty\) , we assume that the lines are given by equations \(x=a\) . Therefore the real projective plane contains every point \((x,y)\) in the coordinate plane plus the ideal line containing one ideal point \(P_m\) for every slope \(m \in \mathbb{R} \cup \{\infty\}\) .


8.1: Introduction to Argument Essay
The idea of an argument often conjures up images of two people yelling and screaming in anger. In writing, however, an argument is very different. An argument is a reasoned opinion supported and explained by evidence. To argue effectively in writing is to advance knowledge and ideas in a positive, persuasive way. Written arguments often fail when they employ ranting rather than reasoning.
In this module, you will develop your skills in argumentative and persuasive writing.
Module Outcomes
After successfully completing this module, you should be able to:
- Determine the purpose and structure of an argumentative essay.
- Understand how to write an argumentative essay.


12.1: The Central Limit Theorem
In the last section, we introduced the normal random variable and primarily discussed how to find the probabilities associated with a normal random variable. We also mentioned that the normal random variable is arguably the most important continuous random variable. We shall now find out why. We first make the following definition:
Definition: Let \( X_1, X_2, X_3, \ldots, X_n \) be random variables. We call the new random variable,
\begin{align*}
\frac{X_1 + X_2 + X_3 + \ldots + X_n}{n}
\end{align*}
the
sample mean
and we denote it by \( \overline{X}_n \).
1) Suppose \(X_i\) models the outcome of our \( i^{th} \) roll of a fair die. Then the random variable \( \dfrac{X_1 + X_2 + \ldots + X_{100}}{100} \) is the sample mean of the 100 rolls.
2) Let us suppose there are 30 people in a classroom and let \(X_i\) models the amount of time person \( i \) spends watching TV on a daily basis. Then the random variable \( \dfrac{X_1 + X_2 + \ldots + X_{30}}{30} \) represents the sample mean.
The question we will consider is if the random variables, \( X_1, X_2, X_3, \ldots, X_n \) are all independent and identically distributed, then can we say anything about their sample mean? The answer is yes and the details are given to us in one of the most important theorems in all of probability and statistics - The Central Limit Theorem.
Theorem: Suppose the random variables \( X_1, X_2, X_3, \ldots, X_n\) are all independent and identically distributed where each random variable has mean \( \mu \) and variance \(\sigma^2\). Then for each fixed \(x\),
\begin{align*}
\lim_{n \rightarrow \infty} P \bigg[ \frac{\overline{X}_n - \mu}{\sigma / \sqrt{n}} \leq x \bigg] = \Phi(x)
\end{align*}
Or equivalently
\begin{align*}
\lim_{n \rightarrow \infty} P \bigg[ \frac{X_1 + X_2 + \ldots + X_n - n\mu}{\sigma \sqrt{n}} \leq x \bigg] = \Phi(x)
\end{align*}
Suppose we have a bunch of random variables, \( X_1, X_2, X_3, \ldots, X_n\) that are all independent and have the same distribution. (For instance, all of them are Uniform random variables with the same parameters or all of them are Exponential random variables with the same parameters). Since they come from the same or identical distribution, then they all have the same average, \( \mu \), and same variance, \( \sigma^2 \). The Central Limit Theorem tells us that:
1) the new random variable, \( \dfrac{X_1 + X_2 + \ldots + X_n}{n} = \overline{X}_n \) will approximately be \( \mathcal{N}(\mu, \frac{\sigma^2}{n}) \).
2) the new random variable, \( X_1 + X_2 + \ldots + X_n \) will be approximately \( \mathcal{N}(n\mu, n \sigma^2) \).
Additionally, notice how general the Central Limit Theorem is! We are saying the distribution of \( X_1, X_2, X_3, \ldots, X_n\) can be anything we wish, as long as the random variables all have the same distribution. Regardless, the sample mean (as well as the sum) will tend to a normal distribution.
Before we consider our examples, allow us to take a few moments to use the following applet. First select a distribution and then select the value of \(n\). Notice that regardless of the starting distribution, the sample mean will tend to a normal distribution as \(n\) increases.
Let \( \overline{X}_{25} \) denote the sample mean of 25 random variables where each random variable comes from a distribution with a mean of 15 and variance of 4. Find \(P(14.4 \leq \overline{X}_{25} \leq 15.6) \).
- Answer
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In this problem \(\mu = 15\) and \( \sigma^2 = 4\). By the Central Limit Theorem, \( \overline{X}_{25} \) will approximately be
\begin{align*}
\mathcal{N} \bigg(\mu, \frac{\sigma^2}{n} \bigg) = \mathcal{N}\bigg(15, \frac{4}{25}\bigg)
\end{align*}To finish the problem, we note that
\begin{align*}
P(14.4 \leq \overline{X}_{25} \leq 15.6) &= normalcdf(-1E99, 15.6, 15, 2/5) - normalcdf(-1E99, 14.4, 15, 2/5) = 0.8663855426
\end{align*}If you do not have a calculator, then you are forced to use the theorem from last section with the CDF chart:
\begin{align*}
P(14.4 \leq \overline{X}_{25} \leq 15.6) &= P \bigg( \frac{14.4-15}{2/5} \leq \frac{X-15}{2/5} \leq \frac{15.6-15}{2/5} \bigg) \\
& \approx P(-1.5 \leq Z \leq 1.5) \\
&= 0.9332 - 0.0668 \\
&= 0.8664
\end{align*}
Let \( \overline{X}_{25} \) denote the sample mean when the underlying random variables come from a distribution whose pdf is given by
\[
f(x) =
\begin{cases}
\frac{x^3}{4} & \text{if} ~ 0 < x < 2 \\
0 & \text{otherwise} \\
\end{cases} \nonumber\ \]
Find/approximate \( P(1.5 \leq \overline{X}_{25} \leq 1.65)\).
- Answer
-
Before we can even answer this question, we have to understand what the problem is asking us. This question is saying suppose you have these random variables, \( X_1, X_2, \ldots X_{25} \) where these random variables all have the above density. We are asked to find or approximate
\begin{align*}
P \bigg(1.5 \leq \frac{X_1 + X_2 + \ldots + X_{25}}{25} \leq 1.65 \bigg)
\end{align*}By the Central Limit Theorem, \( \dfrac{X_1 + X_2 + \ldots + X_{25}}{25} = \overline{X}_{25} \) will approximately be \( \mathcal{N}(\mu, \frac{\sigma^2}{25}) \). Let us find \(\mu \) and \( \sigma^2 \).
\begin{align*}
\mu = \mathbb{E}[X_i] = \int_{all ~x} xf(x) ~ dx = \int_{0}^{2} x \bigg(\frac{x^3}{4} \bigg) ~ dx = \frac{8}{5} = 1.6
\end{align*}\begin{align*}
\sigma^2 = \mathbb{V}ar[X] &= \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \\
&= \mathbb{E}[X^2] - 1.6^2 \\
&= \int_{0}^{2} x^2 \bigg(\frac{x^3}{4} \bigg) ~ dx - 1.6^2 \\
&= \frac{8}{3} - 1.6^2 \\
&= \frac{8}{75}
\end{align*}By the Central Limit Theorem, \( \dfrac{X_1 + X_2 + \ldots + X_{25}}{25} = \overline{X}_{25} \) will approximately be \( \mathcal{N}(\mu, \frac{\sigma^2}{25})\) which turns out to be \( \mathcal{N}(1.6, \frac{8/75}{25}) =\mathcal{N}(1.6, \frac{8}{1875})\).
So \( \overline{X}_{25} \sim \mathcal{N}(1.6, \frac{8}{1875}) \). Hence,
\[ P(1.5 \leq \overline{X}_{25} \leq 1.65) \approx normalcdf(1.5, 1.65, 1.6, \sqrt{\frac{8}{1875}} \approx 0.7151096523 \nonumber\ \]
Or alternatively, if we wish to use the standardization process, we obtain the following:
\begin{align*}
P(1.5 \leq \overline{X}_{25} \leq 1.65) &= P \bigg( \frac{1.5-1.6}{\sqrt{\frac{8}{1875}}} \leq \frac{\overline{X}_{25} -1.6}{\sqrt{\frac{8}{1875}}} \leq \frac{1.65-1.6}{\sqrt{\frac{8}{1875}}} \bigg) \\
&\approx P(-1.531 \leq Z \leq 0.765) \\
&\approx P(-1.53 \leq Z \leq 0.77) \\
&= \Phi(0.77) - \Phi(-1.53) \\
&=0.7794 - 0.0630 \\
&= 0.7164
\end{align*}
Suppose \( X_1, X_2, \ldots, X_{20} \) are all independent and identically distributed as \( \mathcal{U}[0,1] \). Find \(P(X_1 + X_2 + \ldots + X_{20} < 9.1) \).
- Answer
-
Recall if \( X \sim \mathcal{U} [ a , b ] \) then \( \mathbb{E}[X] = \dfrac{a+b}{2} \) and \( \mathbb{V}ar[X] = \dfrac{(b-a)^2}{12} \). Hence, for the \( \mathcal{U}[0,1] \), \( \mathbb{E}[X] = \dfrac{0+1}{2} = \dfrac{1}{2} = \mu \) and \( \mathbb{V}ar[X] = \dfrac{(1-0)^2}{12} = \dfrac{1}{12} = \sigma^2 \).
We know by The Central Limit Theorem, \( X_1 + X_2 + \ldots + X_n \) will be approximately \( \mathcal{N}(n\mu, n \sigma^2) \) and so \( X_1 + X_2 + \ldots + X_{20} \) will be approximately \( \mathcal{N} \big( 20( \frac{1}{2}), 20 (\frac{1}{12}) \big) = \mathcal{N} ( 10, \frac{5}{3} ) \). Thus, we have the following:
\begin{align*} P(X_1 + X_2 + \ldots + X_{20} < 9.1) &= normalcdf \bigg(-1E99, 9.1, 10, \sqrt{\frac{5}{3}} \bigg) \\ & \approx 0.2428584558 \end{align*}


0.2: InfoPage
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This text was compiled on 03/16/2025


<IP_ADDRESS>: Porphyria's Lover
The rain set early in to-night,
The sullen wind was soon awake,
It tore the elm-tops down for spite,
And did its worst to vex the lake:
I listened with heart fit to break.
When glided in Porphyria; straight
She shut the cold out and the storm,
And kneeled and made the cheerless grate
Blaze up, and all the cottage warm;
Which done, she rose, and from her form
Withdrew the dripping cloak and shawl,
And laid her soiled gloves by, untied
Her hat and let the damp hair fall,
And, last, she sat down by my side
And called me.
But passion sometimes would prevail,
Nor could to-night's gay feast restrain
A sudden thought of one so pale
For love of her, and all in vain:
So, she was come through wind and rain.
Be sure I looked up at her eyes
Happy and proud; at last I knew
Porphyria worshipped me; surprise
Made my heart swell, and still it grew
While I debated what to do.
That moment she was mine, mine, fair,
Perfectly pure and good: I found
A thing to do, and all her hair
In one long yellow string I wound
Three times her little throat around,
And strangled her. No pain felt she;
I am quite sure she felt no pain.
As a shut bud that holds a bee,
I warily oped her lids: again
Laughed the blue eyes without a stain.
And I untightened next the tress
About her neck; her cheek once more
Blushed bright beneath my burning kiss:
I propped her head up as before,
Only, this time my shoulder bore
Her head, which droops upon it still:
The smiling rosy little head,
So glad it has its utmost will,
That all it scorned at once is fled,
And I, its love, am gained instead!
Porphyria's love: she guessed not how
Her darling one wish would be heard.
And thus we sit together now,
And all night long we have not stirred,
And yet God has not said a word!


33.1: Animal Form and Function
Skills to Develop
- Describe the various types of body plans that occur in animals
- Describe limits on animal size and shape
- Relate bioenergetics to body size, levels of activity, and the environment
Animals vary in form and function. From a sponge to a worm to a goat, an organism has a distinct body plan that limits its size and shape. Animals’ bodies are also designed to interact with their environments, whether in the deep sea, a rainforest canopy, or the desert. Therefore, a large amount of information about the structure of an organism's body (anatomy) and the function of its cells, tissues and organs (physiology) can be learned by studying that organism's environment.
Body Plans
Animal body plans follow set patterns related to symmetry. They are asymmetrical, radial, or bilateral in form as illustrated in Figure \(\PageIndex{1}\). Asymmetrical animals are animals with no pattern or symmetry; an example of an asymmetrical animal is a sponge. Radial symmetry, as illustrated in Figure \(\PageIndex{1}\), describes when an animal has an up-and-down orientation: any plane cut along its longitudinal axis through the organism produces equal halves, but not a definite right or left side. This plan is found mostly in aquatic animals, especially organisms that attach themselves to a base, like a rock or a boat, and extract their food from the surrounding water as it flows around the organism. Bilateral symmetry is illustrated in the same figure by a goat. The goat also has an upper and lower component to it, but a plane cut from front to back separates the animal into definite right and left sides. Additional terms used when describing positions in the body are anterior (front), posterior (rear), dorsal (toward the back), and ventral (toward the stomach). Bilateral symmetry is found in both land-based and aquatic animals; it enables a high level of mobility.
Limits on Animal Size and Shape
Animals with bilateral symmetry that live in water tend to have a fusiform shape: this is a tubular shaped body that is tapered at both ends. This shape decreases the drag on the body as it moves through water and allows the animal to swim at high speeds. The table below lists the maximum speed of various animals. Certain types of sharks can swim at fifty kilometers an hour and some dolphins at 32 to 40 kilometers per hour. Land animals frequently travel faster, although the tortoise and snail are significantly slower than cheetahs. Another difference in the adaptations of aquatic and land-dwelling organisms is that aquatic organisms are constrained in shape by the forces of drag in the water since water has higher viscosity than air. On the other hand, land-dwelling organisms are constrained mainly by gravity, and drag is relatively unimportant. For example, most adaptations in birds are for gravity not for drag.
| Animal | Speed (kmh) | Speed (mph) |
|---|---|---|
| Cheetah | 113 | 70 |
| Quarter horse | 77 | 48 |
| Fox | 68 | 42 |
| Shortfin mako shark | 50 | 31 |
| Domestic house cat | 48 | 30 |
| Human | 45 | 28 |
| Dolphin | 32–40 | 20–25 |
| Mouse | 13 | 8 |
| Snail | 0.05 | 0.03 |
Most animals have an exoskeleton, including insects, spiders, scorpions, horseshoe crabs, centipedes, and crustaceans. Scientists estimate that, of insects alone, there are over 30 million species on our planet. The exoskeleton is a hard covering or shell that provides benefits to the animal, such as protection against damage from predators and from water loss (for land animals); it also provides for the attachments of muscles.
As the tough and resistant outer cover of an arthropod, the exoskeleton may be constructed of a tough polymer such as chitin and is often biomineralized with materials such as calcium carbonate. This is fused to the animal’s epidermis. Ingrowths of the exoskeleton, called apodemes , function as attachment sites for muscles, similar to tendons in more advanced animals (Figure \(\PageIndex{2}\)). In order to grow, the animal must first synthesize a new exoskeleton underneath the old one and then shed or molt the original covering. This limits the animal’s ability to grow continually, and may limit the individual’s ability to mature if molting does not occur at the proper time. The thickness of the exoskeleton must be increased significantly to accommodate any increase in weight. It is estimated that a doubling of body size increases body weight by a factor of eight. The increasing thickness of the chitin necessary to support this weight limits most animals with an exoskeleton to a relatively small size. The same principles apply to endoskeletons, but they are more efficient because muscles are attached on the outside, making it easier to compensate for increased mass.
An animal with an endoskeleton has its size determined by the amount of skeletal system it needs in order to support the other tissues and the amount of muscle it needs for movement. As the body size increases, both bone and muscle mass increase. The speed achievable by the animal is a balance between its overall size and the bone and muscle that provide support and movement.
Limiting Effects of Diffusion on Size and Development
The exchange of nutrients and wastes between a cell and its watery environment occurs through the process of diffusion. All living cells are bathed in liquid, whether they are in a single-celled organism or a multicellular one. Diffusion is effective over a specific distance and limits the size that an individual cell can attain. If a cell is a single-celled microorganism, such as an amoeba, it can satisfy all of its nutrient and waste needs through diffusion. If the cell is too large, then diffusion is ineffective and the center of the cell does not receive adequate nutrients nor is it able to effectively dispel its waste.
An important concept in understanding how efficient diffusion is as a means of transport is the surface to volume ratio. Recall that any three-dimensional object has a surface area and volume; the ratio of these two quantities is the surface-to-volume ratio. Consider a cell shaped like a perfect sphere: it has a surface area of 4πr 2 , and a volume of (4/3)πr 3 . The surface-to-volume ratio of a sphere is 3/r; as the cell gets bigger, its surface to volume ratio decreases, making diffusion less efficient. The larger the size of the sphere, or animal, the less surface area for diffusion it possesses.
The solution to producing larger organisms is for them to become multicellular. Specialization occurs in complex organisms, allowing cells to become more efficient at doing fewer tasks. For example, circulatory systems bring nutrients and remove waste, while respiratory systems provide oxygen for the cells and remove carbon dioxide from them. Other organ systems have developed further specialization of cells and tissues and efficiently control body functions. Moreover, surface-to-volume ratio applies to other areas of animal development, such as the relationship between muscle mass and cross-sectional surface area in supporting skeletons, and in the relationship between muscle mass and the generation of dissipation of heat.
Animal Bioenergetics
All animals must obtain their energy from food they ingest or absorb. These nutrients are converted to adenosine triphosphate (ATP) for short-term storage and use by all cells. Some animals store energy for slightly longer times as glycogen, and others store energy for much longer times in the form of triglycerides housed in specialized adipose tissues. No energy system is one hundred percent efficient, and an animal’s metabolism produces waste energy in the form of heat. If an animal can conserve that heat and maintain a relatively constant body temperature, it is classified as a warm-blooded animal and called an endotherm . The insulation used to conserve the body heat comes in the forms of fur, fat, or feathers. The absence of insulation in ectothermic animals increases their dependence on the environment for body heat.
The amount of energy expended by an animal over a specific time is called its metabolic rate. The rate is measured variously in joules, calories, or kilocalories (1000 calories). Carbohydrates and proteins contain about 4.5 to 5 kcal/g, and fat contains about 9 kcal/g. Metabolic rate is estimated as the basal metabolic rate (BMR) in endothermic animals at rest and as the standard metabolic rate (SMR) in ectotherms. Human males have a BMR of 1600 to 1800 kcal/day, and human females have a BMR of 1300 to 1500 kcal/day. Even with insulation, endothermal animals require extensive amounts of energy to maintain a constant body temperature. An ectotherm such as an alligator has an SMR of 60 kcal/day.
Energy Requirements Related to Body Size
Smaller endothermic animals have a greater surface area for their mass than larger ones (Figure \(\PageIndex{3}\)). Therefore, smaller animals lose heat at a faster rate than larger animals and require more energy to maintain a constant internal temperature. This results in a smaller endothermic animal having a higher BMR, per body weight, than a larger endothermic animal.
Energy Requirements Related to Levels of Activity
The more active an animal is, the more energy is needed to maintain that activity, and the higher its BMR or SMR. The average daily rate of energy consumption is about two to four times an animal’s BMR or SMR. Humans are more sedentary than most animals and have an average daily rate of only 1.5 times the BMR. The diet of an endothermic animal is determined by its BMR. For example: the type of grasses, leaves, or shrubs that an herbivore eats affects the number of calories that it takes in. The relative caloric content of herbivore foods, in descending order, is tall grasses > legumes > short grasses > forbs (any broad-leaved plant, not a grass) > subshrubs > annuals/biennials.
Energy Requirements Related to Environment
Animals adapt to extremes of temperature or food availability through torpor. Torpor is a process that leads to a decrease in activity and metabolism and allows animals to survive adverse conditions. Torpor can be used by animals for long periods, such as entering a state of hibernation during the winter months, in which case it enables them to maintain a reduced body temperature. During hibernation, ground squirrels can achieve an abdominal temperature of 0° C (32° F), while a bear’s internal temperature is maintained higher at about 37° C (99° F).
If torpor occurs during the summer months with high temperatures and little water, it is called estivation . Some desert animals use this to survive the harshest months of the year. Torpor can occur on a daily basis; this is seen in bats and hummingbirds. While endothermy is limited in smaller animals by surface to volume ratio, some organisms can be smaller and still be endotherms because they employ daily torpor during the part of the day that is coldest. This allows them to conserve energy during the colder parts of the day, when they consume more energy to maintain their body temperature.
Animal Body Planes and Cavities
A standing vertebrate animal can be divided by several planes. A sagittal plane divides the body into right and left portions. A midsagittal plane divides the body exactly in the middle, making two equal right and left halves. A frontal plane (also called a coronal plane) separates the front from the back. A transverse plane (or, horizontal plane) divides the animal into upper and lower portions. This is sometimes called a cross section, and, if the transverse cut is at an angle, it is called an oblique plane. Figure \(\PageIndex{4}\) illustrates these planes on a goat (a four-legged animal) and a human being.
Vertebrate animals have a number of defined body cavities, as illustrated in Figure \(\PageIndex{5}\). Two of these are major cavities that contain smaller cavities within them. The dorsal cavity contains the cranial and the vertebral (or spinal) cavities. The ventral cavity contains the thoracic cavity, which in turn contains the pleural cavity around the lungs and the pericardial cavity, which surrounds the heart. The ventral cavity also contains the abdominopelvic cavity, which can be separated into the abdominal and the pelvic cavities.
 They can work in a variety of settings, although most will have an academic appointment at a university, usually in an anthropology department or a biology, genetics, or zoology department.
Non-academic positions are available in the automotive and aerospace industries where the focus is on human size, shape, and anatomy. Research by these professionals might range from studies of how the human body reacts to car crashes to exploring how to make seats more comfortable. Other non-academic positions can be obtained in museums of natural history, anthropology, archaeology, or science and technology. These positions involve educating students from grade school through graduate school. Physical anthropologists serve as education coordinators, collection managers, writers for museum publications, and as administrators. Zoos employ these professionals, especially if they have an expertise in primate biology; they work in collection management and captive breeding programs for endangered species. Forensic science utilizes physical anthropology expertise in identifying human and animal remains, assisting in determining the cause of death, and for expert testimony in trials.
Summary
Animal bodies come in a variety of sizes and shapes. Limits on animal size and shape include impacts to their movement. Diffusion affects their size and development. Bioenergetics describes how animals use and obtain energy in relation to their body size, activity level, and environment.
Glossary
- apodeme
- ingrowth of an animal’s exoskeleton that functions as an attachment site for muscles
- asymmetrical
- describes animals with no axis of symmetry in their body pattern
- basal metabolic rate (BMR)
- metabolic rate at rest in endothermic animals
- dorsal cavity
- body cavity on the posterior or back portion of an animal; includes the cranial and vertebral cavities
- ectotherm
- animal incapable of maintaining a relatively constant internal body temperature
- endotherm
- animal capable of maintaining a relatively constant internal body temperature
- estivation
- torpor in response to extremely high temperatures and low water availability
- frontal (coronal) plane
- plane cutting through an animal separating the individual into front and back portions
- fusiform
- animal body shape that is tubular and tapered at both ends
- hibernation
- torpor over a long period of time, such as a winter
- midsagittal plane
- plane cutting through an animal separating the individual into even right and left sides
- sagittal plane
- plane cutting through an animal separating the individual into right and left sides
- standard metabolic rate (SMR)
- metabolic rate at rest in ectothermic animals
- torpor
- decrease in activity and metabolism that allows an animal to survive adverse conditions
- transverse (horizontal) plane
- plane cutting through an animal separating the individual into upper and lower portions
- ventral cavity
- body cavity on the anterior or front portion of an animal that includes the thoracic cavities and the abdominopelvic cavities